MTH 337 Scientific and Mathematical Computing

John Feerick

Spring 2017

Prof. Adam Cunningham

A course designed to introduce undergraduate students to fundamental techniques of scientific & mathematical computing through the use of open-ended coding projects (run through python) which seek to model natural environmental and mathematical phenomena.

Projects Included:

Primitive Pythagorean Triples

Floating-Point Numbers

The Mayfly Model

Newton in the Complex Plane

Tatooine

Random Numbers

Statistical Language Models

In [ ]:

Report 1: Primitive Pythagorean Triples

By John Feerick and Marc Roizman

Introduction

      The Pythagorean Theorem is a formula that is taught to students fairly early on in their mathematics careers. It is a fundamental formula for gerometry and has a wide variety of applications from measuring distance between two points to determining the length of a rectangle's diagonal. The general principle of the theorem is, given any right triangle with sides of length $a$ and $b$, and a hypotenuse, or diagonal, of length $c$, the lengths of these sides follow the formula: $$a^2+b^2=c^2$$

      Thus, a pythagorean triple is any triplet of integers $a, b,$ and $c$ which satisfy the Pythagorean Theorem. Once you find one pythagorean triplet however, you can identify infinitely many others simply by multiplying every element of a triplet by some constant $n$. Mathematically speaking, $$a^2+b^2=c^2 \Rightarrow n^2(a^2+b^2)=n^2(c^2) \Rightarrow (na)^2+(nb)^2=(nc)^2$$.

      A Primitive Pythagorean Triple however, is a pythagorean triple for which $a$ and $b$ have only the factor $1$ in common. That is to say, a primitive pythagorean triple is a triplet of positive integers $a,b,$ and $c$ which satisfies $a^2+b^2=c^2$ and for which the greatest common divisor of $a$ and $b$ is $1$.

      In order to calculate the greatest common divisor, one can use The Euclidean Algorithm, which states the following: Assume two integers $a$ and $b$, where $a<b$. The $gcd(a,b)=gcd(a-b,b)$. Therefore, by repeatedly doing this and subtracting the smallest number from the largest, you will eventually reach a point where both your numbers are equal. This final number is the $gcd(a,b)$. To demonstrate an example using the legs of a pythagorean triplet, we will look at $315$ and $572$: $$572, 315 \Rightarrow 572-315, 315 \Rightarrow 257, 315 \Rightarrow 257, 315-257 \Rightarrow 257, 58 \Rightarrow 257-58, 58 \Rightarrow 199, 58 \Rightarrow 141, 58 \Rightarrow 83, 58 \Rightarrow 25, 58 \Rightarrow 25, 33 \Rightarrow 25, 8 \Rightarrow 17, 8 \Rightarrow 9, 8 \Rightarrow 1, 8 \Rightarrow 1, 7 \Rightarrow 1, 6 \Rightarrow 1, 5 \Rightarrow 1, 4 \Rightarrow 1, 3 \Rightarrow 1, 2 \Rightarrow 1, 1$$ Therefore, the $gcd(315,572)$ is $1$, and $(315, 572, 653)$ is a primitive pythagorean triple.

Tasks

      The main objective of this report is to determine how many primitive pythagorean triples lie in the range $a,b \leq 5,000$. In order to do this, a few exercises needed to be completed:

  1. The creation of a function to determine the Greatest Common Divisor of two inputs, using the Euclidian Algorithm. In order to figure out if a pythagorean triplet is primitive, you need to check that the greatest common divisor of $a$ and $b$ is $1$.
  2. The creation of a function to determine whether a number is a perfect square or not, and return True or False accordingly. This will be useful for ensuring that the sum of the squares, $c^2$, is a perfect square itself.
  3. The use of the above two functions to generate a list of primitive pythagorean triples within $a,b \leq 5,000$ and print the total number.
  4. The plotting and analysis of graphs related to these primitive pythagorean triples.

Approach

      The program itself begins with the mygcd function, which performs the Euclidian Algorithm on two inputs and outputs the GCD. It essentially does the same thing humans do when performing the algorithim: while $a$ is not equal to $b$, it uses if statements to subtract the bigger from the smaller number, and eventually returns $a$ when $a$ and $b$ are the same.

      Then comes the is_square function. It's a very simple function, in that it takes an input, and returns True if it's a perfect square, and False if it isn't. It uses the sqrt function to take the square root of the input, then uses the is_integer method to check if the result is an integer. If it is, it returns True, if not, it returns False.

      Finally, the main block of code. We start by creating four lists, one of each for $a, b, c$ and $c^2$. Then we create a for statement for the $1$ - $5000$ range of $a$ (written $1$ - $5001$: ranges in python are inclusive for the first number and exclusive for the second). And another for statement for the $1$ through $a+1$ range of $b$; this ensures duplicates won't be an issue, as $b$ will only go up to the value of $a$. So the way the program will parse the numbers is: $$(a,b) = (1,1), (2,1), (2,2), (3,1),..., (5000,4999), (5000,5000)$$

      Then we obtain a value for $c^2$ by adding $a^2$ and $b^2$. We use an if statement and the mygcd and is_integer functions to verify that the triple is both a pythagorean triple ($a^2+b^2=c^2$) and is primitive ($gcd(a,b)=1$). If both these conditions are met, $a,b,c,$ and $c^2$ are added to their respective lists. Then they are counted up, printed into neat charts, and we can determine that there are $890$ primitive pythagorean triplets for which $a,b \leq 5,000$.

Code and Results

In [1]:
from math import *
In [2]:
#Note: this function is only applicable for a,b greater than 0
def mygcd(a, b):
    "Executes Euclidean Algorithm and returns the Greatest Common Divisor"
    while a!=b :
        if a>b:
            a=a-b
        elif b>a:
            b=b-a
    return a
In [3]:
def is_square(x):
    "Returns True if x is a perfect square"
    root = sqrt(x)
    return root.is_integer()
In [4]:
#Note: there is not an isequal function included because we only use isodd to find the length of the list, 
#not the list itself.The number of evens in a list will equal the length of that list l minus the lenght of isodd(l)
def isodd(l): return [n for n in l if n % 2 != 0] 
"returns a list of all odd numbers in a given list"
In [5]:
def isprime(l):
    "returns a list of prime numbers in a given list"
    primes = []
    for n in l:
        if n==2 or n==3:
            primes.append(n)
            continue
        if n>2 and n%2==0:
            continue
        for i in range(2, int(sqrt(n))+2, 2):
            if n%i==0:
                continue
            if i == int(sqrt(n))+2 or i == int(sqrt(n)):
                primes.append(n)
    return primes
In [22]:
Pythagorean_triples_a = []
Pythagorean_triples_b = []
Pythagorean_triples_c = []
Pythagorean_triples_c_squared = []
for a in range(1, 5001):
    for b in range(1,a):
        c_squared = (a*a)+(b*b)
        if is_square(c_squared) and mygcd(a,b)==1:
            Pythagorean_triples_a.append(a)
            Pythagorean_triples_b.append(b)
            Pythagorean_triples_c.append(sqrt(c_squared))
            Pythagorean_triples_c_squared.append(c_squared)    
print("There are",len(Pythagorean_triples_a),"Pythagorean triples between 1 and 5000")

There are 890 Pythagorean triples between 1 and 5000
In [23]:
%matplotlib inline
#Calls show automatically when each cell is executed
In [24]:
import matplotlib.pyplot as plt
In [25]:
#Plot of only primitive pythagorean triplets in the range
x=Pythagorean_triples_a
y=Pythagorean_triples_b
plt.plot(x, y, 'r^', ms=4)
plt.xlabel('a')
plt.ylabel('b')
plt.title('Plot of Primitive Pythagorean Triples')
Out[25]:
Text(0.5, 1.0, 'Plot of Primitive Pythagorean Triples')
In [27]:
#Generates doubles
import math
Pythagorean_triples_a2 = []
Pythagorean_triples_b2 = []
Pythagorean_triples_c2 = []
Pythagorean_triples_c2_squared = []
for a in range(1, 5001):
    for b in range(1, 5001):
        c_squared = (a*a)+(b*b)
        if is_square(c_squared) and mygcd(a,b)==1:
            Pythagorean_triples_a2.append(a)
            Pythagorean_triples_b2.append(b)
            Pythagorean_triples_c2.append(sqrt(c_squared))
            Pythagorean_triples_c2_squared.append(c_squared)    
print("There are",len(Pythagorean_triples_a2)/2,"Pythagorean triples between 1 and 5000")

There are 890.0 Pythagorean triples between 1 and 5000
In [11]:
x=Pythagorean_triples_a2
y=Pythagorean_triples_b2
plt.plot(x, y, '.', ms=4)
plt.xlabel('a')
plt.ylabel('b')
plt.title('Plot of Mirrored Primitive Pythagorean Triples')
Out[11]:
Text(0.5, 1.0, 'Plot of Mirrored Primitive Pythagorean Triples')
In [28]:
import math
Pythagorean_triples_a1 = []
Pythagorean_triples_b1 = []
Pythagorean_triples_c1 = []
for a in range(1, 5001):
    for b in range(1, 5001):
        c_squared = (a*a)+(b*b)
        if is_square(c_squared):
            Pythagorean_triples_a1.append(a)
            Pythagorean_triples_b1.append(b)
            Pythagorean_triples_c1.append(sqrt(c_squared))
print("There are",len(Pythagorean_triples_a1)/2,"Non-Primitive Pythagorean triples between 1 and 5000")

There are 6616.0 Non-Primitive Pythagorean triples between 1 and 5000
In [13]:
#Plot for all Pythagorean triples in the range
x=Pythagorean_triples_a1
y=Pythagorean_triples_b1
plt.plot(x, y, '.', ms=4)
plt.xlabel('a')
plt.ylabel('b')
plt.title('Plot of All Pythagorean Triples')
Out[13]:
Text(0.5, 1.0, 'Plot of All Pythagorean Triples')
In [14]:
x=list(range(1,891))
Pythagorean_triples_c_squared.sort()
y=Pythagorean_triples_c_squared
plt.plot(x, y, 'r-', ms=4)
plt.xlabel('Ordinal Position of c^2 value')
plt.ylabel('c')
plt.title('c^2 Values for Primitive Pythagorean Triples')
Out[14]:
Text(0.5, 1.0, 'c^2 Values for Primitive Pythagorean Triples')
In [15]:
len(isprime(Pythagorean_triples_c))
Out[15]:
449
In [16]:
len(isodd(Pythagorean_triples_c))
Out[16]:
890
In [17]:
len(isodd(Pythagorean_triples_a))
Out[17]:
444
In [18]:
len(isprime(Pythagorean_triples_a))
Out[18]:
218
In [19]:
len(isprime(Pythagorean_triples_b))
Out[19]:
218
In [20]:
len(isodd(Pythagorean_triples_b))
Out[20]:
446

Interpretation

      After generating and plotting $a$ and $b$ from all of the primitive triples, we immediately realized that we hadn't accounted for there being doubles of each pair. Despite our efforts we were not able to find a method to plot unique pairs of $a$ and $b$ that didn't have either $a > b$ or $a < b$. Upon inspection the graph seems to experience some parabolic patterns extending in the positive directions of $a$ and $b$ from the origin. We then plotted the doubles which further exposes the patterns, with three radiating concentric paraboli seeming to be spiraling around the origin. In an attempt to see more manifest, we decided to generate all pythagorean triples with $a, b < 5000$ including non-primitives. Once plotted, the parabolic patterns remain consistent but there are now multiple linear lines eminating from the origin; this is the result of pairs having common divisors.

      We next decided to focus on generating data for $c$'s fluctuations. Initially, we simply plotted the all of the $c$ values against the range of the number of $c$ values, but did not extract any meaningful data from that besides $c$ generally increasing with $a$ and $b$, but since $b$ values were constantly reduced the $c$ values were as well. We utilized the sort() method to order the $c$ values based on their increasing size and then plotted that. The $c$ values appear linear until roughly $800$, when they take an upwards quadratic turn, and the squared values form a consistently quadratic line with what appears to be taking a sharper turn upwards at $x = 890$.

      Finally, we were curious to explore what other discrete patterns could be found in the triples. After writing and applying functions which returned lists of the odd and prime numbers of the triples, we observed some interesting results. Over half of the values for $c$ are prime, and all of them are odd. Additionally, there are $218$ prime values in both the lists for a and b and close to half of each are even and half odd; interestingly, there are $2$ more odd numbers in $b$ than there are in $a$. Every pair is comprised of one even and one odd number. While we are unsure of how to interpret or prove these results, we found them to be noteworthy.

MTH337 Report 2: Floating Point Numbers

Introduction

    While most take our system of describing numbers in base 10 (or decimal) for granted, it is only one of infinite methods for representing numberical quantities. Interestingly, the cause of this is rooted in anthropology. It is believed that most human civiliations counted in base 10 due to humans having 10 fingers; the next most commonly used system is base 20, acconting for 10 fingers and toes. However, our computers preform calculations using the base 2 (or binary) system because the language of computers is built around switches which are either on or off. Even though there are only two possibly values a computer could represent for each digit, once you begin to create different types of gates the possibilities quickly grow. For this report we are exploring in turn the mathematical limitations that using the binary system to represent the real numbers causes.
    The IEEE754 is a system used for approximating non-integers. We use it to convert from binary to decimal while simultaneously allowing non-whole decimal numbers. Instead of interpreting a binary string as a simple base 2 representation, it is read differently to expand to the real numbers. The following is the standard formula for converting the computer representation of a number into an actual binary number:
$${-1}^s * (1.mantissa) * {2}^{exp-bias}$$ 
    The mantissa refers to the binary string that is after 1, and the bias is a fixed integer that is used to more easily represent very large and very small numbers by changing the exponent accordingly. For this report, I am attempting to find the bias and mantissa for various floating point numbers that python uses. Additionally, I am going to find the value of machine epsilon, which is a measure of the relative space between numbers that are successive. Finally, I will use a taylor series approximation to display the accuracy of floating point representations.

Tasks Stated/Approach

In [1]:
#This is a very simple function that we went through in class, the logic behind it is that we keep decaying the variable to a 
#smaller power of 2, and once adding that value to 1 produces a result of 1, 1-n will give us the base 2 exponent of machine 
#epsilon, and 2 raised to 1-n will give us machine epsilon's value. This is because machine epsilon is the value 
#between two successive numbers, so if we were to add a number smaller than machine epsilon to one, the result will be one.
def find_epsilon():
    "returns the value of machine epsilon, or the smallest value larger than one which can be represented by a floating point as well as its base 2 exponent"
    n = 1
    x = 2**(-n)
    while 1 + x !=1:
        n+=1
        x=2**(-n)
    print('The value of machine epsilon is',2**(1-n),'and its base 2 exponent is',1-n)
In [2]:
find_epsilon()

The value of machine epsilon is 2.220446049250313e-16 and its base 2 exponent is -52
In [9]:
#If we try to represent a number larger than the largest floating point number python can store, it will result in an error. This
#function exploits that by using a try except statement to exit a while loop which iterates up powers of 2 once that error occurs, 
#However, this number is not actually the largest floating point, but rather the maximum power of 2 the IEEE754 can represent.
#To find the maximum value, we must then continually add decreasing powers of 2 so that all digits in the mantissa
#are equal to 1. To do this I ran another while loop that executed until adding any more would result in infinity. This number
#multiplied by 2^(exp-bias) will result in the largest representable floating point number.
def find_largest():
    "returns the largest floating point number that can be represented"
    n = 1.
    x = 2**n
    try:
        while x != 2**(n-1):
            n +=1
            x = 2**n
    except:
        exp = n - 1
        n = n-1
        exp = n
        x = 2**(n)
        while n > 0 and x + 2**(n-1) != float("inf"):
            x += 2**(n-1)
            n -= 1
        print(x,'is the largest representable floating point and',exp,'is its base 2 exponent')
In [10]:
find_largest()

1.7976931348623157e+308 is the largest representable floating point and 1023.0 is its base 2 exponent
In [11]:
#This is a very similar function to find_largest() but is slightly simpler. If I we iterate though powers of 2 until python 
#represents it as 0.0, we know that whichever power came before is the smallest base 2 power representable. The value of that 
#power of 2 will actually equal the smallest value representabe in python, since in its binary representation all preceeding 
#digits are 0. Attempting to subtract anything from that number will result in 0, since that would be outside of the bit space
#allocated for the mantissa.
def find_smallest():
    "Returns the smallest float Python is able to represent"
    n = 0.
    x = 2**n
    while x != 0.0:
        n -=1
        x = 2**n
    n = n + 1
    x = 2**(n)
    print(x,'is the smallest representable floating point and',n,'is its base 2 exponent')
In [12]:
find_smallest()

5e-324 is the smallest representable floating point and -1075.0 is its base 2 exponent

Table

We are finding the smallest floats, not the least, so all signs are going to be 0. The tables for biggest and smallest float should be obvious as they were determined by the above functions. The exponent of 1 + machine epsilon is going to be 0 and the mantissa is going to be all 0's except for one at the end, which by itself is equal to machine epsilon. The smallest normalized number would have all 0's in the mantissa and have the same exponent as the smallest possible float, and is literally just 1 to that power. Denormalization is used since the range of exponents between the largest and smallest floats is greater than can be stored using the exponent bits. The extra exponents arise from the zeros after the decimal point in the mantissa when using denormalized numbers (0.1, 0.01, 0.001 etc). The number of bits reserved for the mantissa is 52, the absolute value of the exponent for machine epsilon. Since that represents the smallest interval between successive numbers, its exponent shows have long the mantissa is.  The value reserved for the exponent is going to equal 11 bits, coming from the fact that this has been tested on a 64-bit system and 64 - 52 (Mantissa) - 1 (Sign bit) = 11.
Number Sign Exponent Mantissa Value
Biggest Float 0 1111...1111 1111...1111 1.7976931348623157e+308
1 + $\epsilon$ 0 Bias 0000...0001 1 + 2.220446049250313e-16
Smallest Normalized 0 0000...0001 0000...0000 1e-75
"First" Denormalized 0 0000...0000 0000...0001 $2^{-52} * 2^{-51}$
Smallest Float 0 -1111...1111 0000...0001 5e-324
In [1]:
#The graph below is plotting the function log(1 + x)/x for values of x between -1 and 1. This graph highlights the lack of
#precision that floats can have, as we are only able to represent numbers in intervals of epsilon. Numbers which are inbetween
#these intervals get truncated based on what they round to, which accounts for the saw-like waves. As x approaches 0 the function 
#will begin to act like a hyberbola (y = constant/x) with asymptotes at x = 0.
%pylab inline
x = linspace(-1e-10,1e-10,200)
y = np.where(x==0, 1.0, np.log(1+x)/x)
plt.xlabel('x')
plt.ylabel('floating point approximation of log(1 + x)/x')
plot(x,y,'b',alpha=.5)

Populating the interactive namespace from numpy and matplotlib
Out[1]:
[<matplotlib.lines.Line2D at 0x7f399a7d3c18>]
In [14]:
#Here I continued the taylor series approximation we went over in class to another term. I won't include the algebra, but the 
#series for f(x) continues in its pattern of the sum of x^n/n (n = 1,2,3...). g(x) is simply that series with every term divided
#by x. It is plotted in red while the original graph is blue. The taylor series approximation doesn't account for the lack of a 
#y-value at x = 0, but otherwise gives a better depiction of what this slice of the graph should look like. When the floating 
#point numbers begin to get very small, theirapproximations become more inaccurate whereas the taylor series is constant besides
#slight hyperbolic behavior at x = 0.
plt.figure(figsize=(10, 6))
end = 2.0e-10
x = linspace(-1e-10,1e-10,200)
y = np.where(x==0, 1.0, np.log(1+x)/x)
p = linspace(-1e-10,1e-10,200)
q = 1 - p/2 + p**2/3 - p**3/4
plt.xlabel('x')
plt.ylabel('log(1 + x)/x')
plt.plot(x, y, 'b',alpha=.5)
plt.plot(p, q,'r',alpha=.5)
Out[14]:
[<matplotlib.lines.Line2D at 0xd9453dbd30>]

Exercise 7 is completed algebraically. The formula for error in a taylor series with n terms is $$\frac{f^{n+1}(\xi)}{(n + 1)!} * x^{n + 1}$$ So for a series with 4 terms and $\xi$ set equal to 0 (to find the maximum error) we find that the largest possible value for the error is $\frac{x^4}{5}$

We set that equal to $10^{-16}$ and solve for x to find x = 0.00026591479.

Interpretation

    Exercise 1 was very simple to complete. Exercise 2 was slightly more difficult. I erroneously did not include the 2nd while loop in it, and upon googling the maximum float in IEEE754 knew my results were wrong. I'm unsure if that counts as cheating, but I found such information to be critical as a lot of the logic behind this report was pretty challenging for me. I was then puzzled by exercise 3, but I had used the same structure for the code as I had with exercise 2 and kept running into problems. After thinking about it like binary strings I realized that I actually was generating the smallest value just from finding the smallest power of 2, and whenever I subtracted from it I'd get 0. 
    Setting up the table in exercise 4 was pretty tricky. I first assumed that the smallest float was equal to the "first" denormalized but once I paid closer attention to the formula used for denormalized numbers I realized I was incorrect. Realizing that the bias was 52 made things much clearer. I was also unsure what value to use for the exponent, either the exp in the IEEE754 formula or that minus the bias, and only upon looking at the table on the report website did I realize which to use.
    The result of the taylor series approximation wasn't very surprising to me. I found it interesting that in the floating point representation the frequency of the waves decreases and amplitude increases as x becomes positive. I've tried thinking it over, but wasn't able to come up with anything concrete in time. I assume it relates to the bias. The result of exercise 7 didn't seem to be of too much consequence to me. I tried applying it but didn't have the time to find anything meaningful.
In [ ]:

Report 3

Introduction

    For this report we wanted to observe the behavior of a certain discrete dynamical system, the Mayfly Model. This model approximates a year's population of mayfly butterflies based on the population of the previous year and a certain growth factor. What is interesting is that the level of effect a given initial population has over time, which one may expect to be great, is based upon the growth factor. In class we defined a function to find the population to be
$$x_{t+1} = b * x_t * (1 - x_t)$$ 
where x is the population, t is the number of years since the initial population and b is the growth factor.
    Now that we have an equation representing the model, we can determine if there is any behavior displaying aspects of the butterfly effect. In order to do so, we want to observe the behavior of population growth during some specified range of time where we expect the population to have converged for all values of b on some interval; for this report we choose that interval to be [1,4]. If the population converges to some value at b, we know that growth factor does not lead to chaotic behavior. Otherwise, it does.
    Once we know what values of b lead to chaotic behavior, we can graph the separation of trajectories for very close starting populations to see how greatly different results can be caused. We then can analyze such graphs to determine a line of best fit to calculate the separate trajectories.

Tasks Stated/Approach/Results/Interpretation

Exercise 1

For this exercise we had to run the mayfly model for different values of b and plot the change for some length of time to determine if the population ever converges. In this mayfly function the variables in the above equation are taken to be the arguments. Note that no matter what t is supplied, we run the model 1000 times before recording its value. This is because based on values of x and b, the population converges at different times (that is, if it converges). All populations will have converged by t = 1000.
In [3]:
import numpy as np
import matplotlib.pyplot as plt
def mayfly(x,b,t):
    "MAYFLY"
    x_values = np.empty(t)     #Initial Population at t = 0
    print(x)
    for i in range(1,t+1001):
        x = b*(1-x)*x
        if i>1000:
            x_values[i-1001] = x
    return x_values
In [33]:
plt.plot(mayfly(.3,1.2,100))
plt.xlabel('t - 1000 years')
plt.ylabel('population')

0.3
Out[33]:
<matplotlib.text.Text at 0xa5e2037a90>
In [34]:
plt.plot(mayfly(.3,2.0,100))
plt.xlabel('t - 1000 years')
plt.ylabel('population')

0.3
Out[34]:
<matplotlib.text.Text at 0xa5e2097f98>
In [35]:
plt.plot(mayfly(.3,3.0,100))
plt.xlabel('t - 1000 years')
plt.ylabel('population')

0.3
Out[35]:
<matplotlib.text.Text at 0xa5e20fb080>
In [36]:
plt.plot(mayfly(.3,3.9,100))
plt.xlabel('t - 1000 years')
plt.ylabel('population')

0.3
Out[36]:
<matplotlib.text.Text at 0xa5e215f1d0>
In [37]:
plt.plot(mayfly(.7,3.9,100))
plt.xlabel('t - 1000 years')
plt.ylabel('population')

0.7
Out[37]:
<matplotlib.text.Text at 0xa5e21b9c88>
The above graphs were chosen since they detail the model's behavior pretty well. I feel as though generating more would be redundant, but if I were to it would quickly become observable that for all b in [1,3), the population will converge by t = 1000. However, for all values greater than or equal to b the population continues to oscillate. At values closer to 3 these oscillations have peaks that are equal to the amplitude, but at larger growth rates the oscillations become a bit more complex and have a larger period, such as in the graph above (whose period is longer than the graph's x-axis).

Exercise 2

For exercise 2 we had to plot what x converges to asymptotically over a range of time of our choosing for all b in [1,4]. I chose tmin to be 1000 and tmax to be 1400 just because I enjoyed the plots it produced the most. We were told to use x = .5 for all values of b. However, the same graph is produced regardless of the stating population, meaning that the convergence of the populaion is only dependent upon b and not x.
In [5]:
b = np.linspace(1,4,10000)
x = np.ones_like(b)*0.5 
for i in range(1000):
    x = b*(1-x)*x
for i in range(300):
    x = b*(1-x)*x
    i += 1
    x_values = x
    plt.plot(b,x_values,'b.',ms=.1,alpha=.3)
    plt.xlabel('b')
    plt.ylabel('Where the population converges to')

Since this plot begins to exhibit non-convergent behavior at b = 3.0, I decided to create another plot for b in [3,4].
In [7]:
b = np.linspace(1,4,10000)
x = np.empty(len(b))
for i in range (0,len(x)):
    x[i] = .5
i = 0
while i < 1000:
    x = b*(1-x)*x
    i += 1
while i < 1300:
    x = b*(1-x)*x
    i += 1
    x_values = x
    plt.plot(b,x_values,'b.',ms=.1,alpha=.3)
    plt.xlabel('b')
    plt.ylabel('Where the population converges to')
plt.xlim(3,4.0)
Out[7]:
(3, 4.0)
While there was definitely some visible pattern in the first plot I generated, the above graph does a much better job of visualizing this. As the population begins to oscillate with increasing b, sub-oscillations begin to develop, eventually leading to the sum of sinusoidal patterns before us. This graph provides some insight into exercise 1, where any values above b = 3 led to increasing populations not converging, and at higher values of b the population change became even more unpredictable. I decided to plot the graph again, but this time from b in [3.5,4] and with tmax = 2000 to observe any further patterns.
In [64]:
b = linspace(1,4,10000)
x = np.empty(len(b))
for i in range (0,len(x)):
    x[i] = .5
i = 0
while i < 2000:
    x = b*(1-x)*x
    i += 1
while i < 3000:
    x = b*(1-x)*x
    i += 1
    x_values = x
    plt.plot(b,x_values,'b.',ms=.1,alpha=.3)
    plt.xlabel('b')
    plt.ylabel('Where the population converges to')
plt.xlim(3.5,4)
Out[64]:
(3.5, 4)
There's obviously quite a bit going on in this graph. As I added more points more sinusoidal patterns become clear. There's a greater likelihood of populations appearing on the sinusoidal curves that appear to origniate from the curves in which x is just oscillating. In some areas, the populations nearly converge again, but then scatter once again.
The above graph supplies more than enough evidence for the mayfly model exhibiting the butterfly effect. Even though at high values of b there are still patterns of convergence that would allow us to possibly estimate x at some given time, the behavior becomes far too randomized. Within the range at point b, with enough time the populations will hit almost every value. Just altering b ever so slightly has given us very, very different and unpredictable results.

Exercise 3

Now we want to find the trajectories of two very close initial populations and then plot their difference each year to find how the separation grows or decays. I defined the following function to take an initial population and growth factor as arguments. I then create another initial population very close to the argument and plot the absolute value of their difference for each year. For this exercise we use a relatively small and low range of value for t since there is no separation once the populations converge. I set the function to return the array of separated values so that I could easily use them in the next exercise.
In [66]:
def population_separation(m, b):
    "Gives a plot of the difference in trajectories for different initial populations"
    s = .00000000001
    n = m + s
    s_values = np.empty(30)
    s_values[0] = s
    j = 1
    while j < 30:
        m = b*(1-m)*m
        n = b*(1-n)*n
        s_values[j] = abs(m - n)
        j += 1
    return s_values

I then plotted the separation values with various arguments for m and b against a logarithmic y-axis using the semilogy function. We expect there to be some value of b for which the popuation grows exponentially over time. To monitor this I want to plot the logarithmic separation for a variety of x and b values. For the sake of brevity and style, I used four of each as can be seen in the cell below.
In [117]:
rows, cols = 4, 4
x_values = [.2,.4,.6,.8]
b_values = [1.5,2.5,3.5,4]
t = np.linspace(0, 30, 30)
i = 1
plt.figure(figsize=(15,15))
for b in b_values:
    for x in x_values:
        plt.subplot(rows, cols, i)
        plt.semilogy(population_separation(x,b))
        i += 1
        if x == .2:
            plt.ylabel(b)
        if b == 4:
            plt.xlabel(x)
plt.suptitle("Logarithmic separation (y-axis) for varying values of x (bottom) against varying values of b (left) at time t (x-axis)")
Out[117]:
<matplotlib.text.Text at 0xa58a366ba8>

Exercise 4

Finally, we want to find a region where the separations are growing exponentially which can be described by this equation
$$log(s(t)) = log(s(0)) + r * t = w_0 + w_1 * t$$ 
That equation relates the formula to find the line of least regression with the equation we are trying to satisfy. Such a region would have a linear slope on some region of t. Examining the above graphs, I chose to find the values of $w_0$ and $w_1$ for x = .8 and b = 1.5. To find the values we solve the following matrix equation:

title

Below I solve for $w_0$ and $w_1$.
In [102]:
x = arange(5,14)
s = population_separation(.8,1.5)
y = log(s[5:14])
xbar = np.mean(x)
xbar2 = np.mean(x**2)
ybar = np.mean(y)
xybar = np.mean(x*y)
A = [[1,xbar],[xbar,xbar2]]
b = [ybar,xybar]
w0, w1 = np.linalg.solve(A,b)
Out[102]:
array([-23.70553173,  -0.68046761])
Therefore our equation becomes
$$log(s(t)) = log(s(0)) + r * t = -23.70553173 - 0.68046761 * t$$ 
for x = .8 and b = 1.5, and this is the result of plotting it on the same graph of the trajectories from t = 5 to t = 14:
In [116]:
x = arange(5,14)
y = w0 + w1*x 
semilogy(exp(y),"--")

n = population_separation(.8,1.5)
s = n[5:14]
semilogy(s)
Out[116]:
[<matplotlib.lines.Line2D at 0xa589547dd8>]

The two graphs are identical, meaning that I succesfully found the equation for the linear least squares.

Report 4

Introduction

Complex numbers are defined to be a pair of 2 numbers of the form $x + yi$, where x and y are both real numbers and $i^2 = -1$. Even though they themselves are not a part of the set of "real numbers" (the one dimensional number line), they still are of great value as they are used frequently in physics and their study in mathematics allows properties that the real numbers don't have but the complex do to be discussed. These numbers can be represented on the complex plane x vs y (where x and y make up an imaginary number) using the same polar coordinates that we can use for the real's, with r being the distance of the point from the real axis and theta the angle that r forms with the real axis.

Roots of unity for some integer n are complex numbers which when raised to n evaluate to 1. For this report we are considering the cubic roos of unity such that $z^3 = 1$. The roots in the complex plane are given by $r = 1$ and $\theta = 0, \frac{2\pi}{3}$, and $\frac{4\pi}{3}$. The points in the complex plane are all going to converge to one of these roots which we can represent visually. Additionally, we can also represent the rate at which points converge to a root by repeating iterations of the cubic equation, which we are going to essentially treat as a recursive sequence.

Tasks Stated/Approach/Results/Interpretation

Exercise 1

This exercise is pretty cut and dry, but I find getting the arrays right to be somewhat tricky. First, we generate an array of the three complex roots of the equation by transforming the polar coordinates to the complex form. Now that we have our imaginary roots, we generate a complex plane with 500 points on both the x and y axes with both between -1 and 1. To create the plane, we make two 2-D arrays using the numpy meshgrid function and then add those together. To make all of the points converge to a root, we simply apply Newton's Method at every different point and see what root it converges to by checking using a tolerance and creating three separate arrays which contain all the values that converge to a specific root.

In [1]:
import warnings
warnings.filterwarnings('ignore')
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
In [2]:
roots = np.array([complex(np.cos(2*np.pi*i/3),np.sin(2*np.pi*i/3)) for i in range(3)])
npts = 500
x = np.linspace(-1.,1.,npts)
y = np.linspace(-1.j,1.j,npts)
X,Y = np.meshgrid(x,y)
Z = X + Y
niters = 30
for i in range(niters):
    Z -= (Z**3 - 1)/(3*Z**2)
tol = .0001
root1 = np.abs(Z - roots[0]) < tol
root2 = np.abs(Z - roots[1]) < tol
root3 = np.abs(Z - roots[2]) < tol

Exercise 2

Now we merely seek to actually graph these regions. We create an empty array that can hold all 500x500 points and also each value for color, which takes three different arguments to work with the imshow function. Since we only have three regions that which for each point only converge to one root, we simply set the three color values to the root arrays themselves, since for every index there will be two 0's and one 1.

In [3]:
img = np.zeros([npts,npts,3])
img[:,:,0] = root1
img[:,:,1] = root2
img[:,:,2] = root3
plt.figure(figsize=(8,8))
plt.imshow(img)
Out[3]:
<matplotlib.image.AxesImage at 0xfc00d65550>

Exercise 3

We just want to zoom in on an interesing region of the complex plain. In all honesty I thought that zooming in on the origin was the only plot which is of interest, the rest aren't really going to give much better of a perspective. To do this, we just change the bounds in the linspace functions which we use to define the variables x and y. Once the basins of attraction appear to form 6 straight lines, any further zooming will always produce the same image if you keep increasing the number of iterations that we use. This shows that this pattern carries on infentismally long, seen below in two plots with the second's domains being a millionth the size of the first. It seems as though Newton's method is quite thorough for determining roots.

In [4]:
x = np.linspace(-.1,.1,npts)
y = np.linspace(-.1j,.1j,npts)
X,Y = np.meshgrid(x,y)
Z = X + Y
niters = 300
for i in range(niters):
    Z -= (Z**3 - 1)/(3*Z**2)
tol = .0001
root1 = np.abs(Z - roots[0]) < tol
root2 = np.abs(Z - roots[1]) < tol
root3 = np.abs(Z - roots[2]) < tol
img = np.zeros([npts,npts,3])
img[:,:,0] = root1
img[:,:,1] = root2
img[:,:,2] = root3
plt.figure(figsize=(8,8))
plt.imshow(img)
Out[4]:
<matplotlib.image.AxesImage at 0xfc0002a080>
In [5]:
x = np.linspace(-.00000001,.00000001,npts)
y = np.linspace(-.00000001j,.00000001j,npts)
X,Y = np.meshgrid(x,y)
Z = X + Y
niters = 300
for i in range(niters):
    Z -= (Z**3 - 1)/(3*Z**2)
tol = .0001
root1 = np.abs(Z - roots[0]) < tol
root2 = np.abs(Z - roots[1]) < tol
root3 = np.abs(Z - roots[2]) < tol
img = np.zeros([npts,npts,3])
img[:,:,0] = root1
img[:,:,1] = root2
img[:,:,2] = root3
plt.figure(figsize=(8,8))
plt.imshow(img)
Out[5]:
<matplotlib.image.AxesImage at 0xfc00821c18>

Exercise 4

As I stated above, for certain points you need to complete more iterations of Newton's method to find a point we can say it converges to. As we approach basins of attraction, the number of iterations to find roots decreases to 0. To visualize this we create a 2-D array with length and width 500 for which we set each index equal to the amount of times that an iteration converges to a root for all 500 points. We then divide that number by the total maximum number of iterations (there is no actual maximum as points which are to scale it between 0 and 1, which we then use as an argument in the imshow function that determines the color's transparency. In the plot below, the areas which are white take the converge immediately and crop up around the origin and other bases of attraction.

In [6]:
roots = np.array([complex(np.cos(2*np.pi*i/3),np.sin(2*np.pi*i/3)) for i in range(3)])
npts = 500
x = np.linspace(-1.,1.,npts)
y = np.linspace(-1.j,1.j,npts)
X,Y = np.meshgrid(x,y)
Z = X + Y
niters = 30
intensity = np.zeros((npts,npts))
tol = .0001
for i in range(niters):
    Z = Z - (Z**3 - 1)/(3*Z**2)
    for index in range(3):
        intensity += np.abs(Z - roots[index]) < tol
root1 = np.abs(Z - roots[0]) < tol
root2 = np.abs(Z - roots[1]) < tol
root3 = np.abs(Z - roots[2]) < tol
intensity /= niters
img = np.zeros([npts,npts,4])
img[:,:,0] = root1
img[:,:,1] = root2
img[:,:,2] = root3
img[:,:,3] = intensity
plt.figure(figsize=(8,8))
plt.imshow(img)
Out[6]:
<matplotlib.image.AxesImage at 0xfc00885a58>

There are other ways to scale the intensity variable so that it remains between 0 and 1 but produces different values. I've included some examples of these below.

In [7]:
#This is essentially the same plot except that points for which the maximum number of iterations are required converge 
#immediately, since that's the only thing taking mod 1 would change for values between 0 and 1.
Z = X + Y
intensity = np.zeros((npts,npts))
for i in range(niters):
    Z = Z - (Z**3 - 1)/(3*Z**2)
    for index in range(3):
        intensity += np.abs(Z - roots[index]) < tol
intensity /= niters
intensity %= 1
img[:,:,3] = intensity
plt.figure(figsize=(8,8))
plt.imshow(img)
Out[7]:
<matplotlib.image.AxesImage at 0xfc0062b048>
In [8]:
#Points should now be converging similarly to the previous function, except that we essentially made it repeat with a period of 
#2, as can be seen along the bases of attraction.
Z = X + Y
intensity = np.zeros((npts,npts))
for i in range(niters):
    Z = Z - (Z**3 - 1)/(3*Z**2)
    for index in range(3):
        intensity += np.abs(Z - roots[index]) < tol
intensity /= niters
intensity *= 2
intensity %= 2
img[:,:,3] = intensity
plt.figure(figsize=(8,8))
plt.imshow(img)
Out[8]:
<matplotlib.image.AxesImage at 0xfc00944ba8>
In [9]:
#Using the squares of intensity and iterations causes points furthest from the basis of attraction, where the intensity is 1, to 
#converge more slowly and those closer to the bases of attraction to converge more quickly.
Z = X + Y
intensity = np.zeros((npts,npts))
for i in range(niters):
    Z = Z - (Z**3 - 1)/(3*Z**2)
    for index in range(3):
        intensity += np.abs(Z - roots[index]) < tol
intensity = intensity**2/niters**2
img[:,:,3] = intensity
plt.figure(figsize=(8,8))
plt.imshow(img)
Out[9]:
<matplotlib.image.AxesImage at 0xfc015675c0>
In [10]:
#Taking the roots shows points taking much longr to converge.
Z = X + Y
intensity = np.zeros((npts,npts))
for i in range(niters):
    Z = Z - (Z**3 - 1)/(3*Z**2)
    for index in range(3):
        intensity += np.abs(Z - roots[index]) < tol
intensity = np.sqrt(intensity)/np.sqrt(niters)
img[:,:,3] = intensity
plt.figure(figsize=(8,8))
plt.imshow(img)
Out[10]:
<matplotlib.image.AxesImage at 0xfc015c7048>
In [11]:
#Taking sin and cosine of intensity displays obviously periodic beavior.
Z = X + Y
intensity = np.zeros((npts,npts))
for i in range(niters):
    Z = Z - (Z**3 - 1)/(3*Z**2)
    for index in range(3):
        intensity += np.abs(Z - roots[index]) < tol
intensity = np.sin(intensity)
img[:,:,3] = intensity
plt.figure(figsize=(8,8))
plt.imshow(img)
Out[11]:
<matplotlib.image.AxesImage at 0xfc03150908>
In [12]:
Z = X + Y
intensity = np.zeros((npts,npts))
for i in range(niters):
    Z = Z - (Z**3 - 1)/(3*Z**2)
    for index in range(3):
        intensity += np.abs(Z - roots[index]) < tol
intensity = np.cos(intensity)
img[:,:,3] = intensity
plt.figure(figsize=(8,8))
plt.imshow(img)
Out[12]:
<matplotlib.image.AxesImage at 0xfc031b0320>

After some independent research, I decided to create Newton fractals in the complex plane for different roots of unity. Below are plots describing the convergence of solutions in the complex plane for $z^n = 1$ for some natural number n.

In [13]:
roots = np.array([-complex(np.cos(np.pi*i/2),np.sin(np.pi*i/2)) for i in range(4)])
npts = 500
x = np.linspace(-1.,1.,npts)
y = np.linspace(-1.j,1.j,npts)
X,Y = np.meshgrid(x,y)
Z = X + Y
niters = 100
for i in range(niters):
    Z -= (Z**4 - 1)/(4*Z**3)
tol = .01
root1 = np.abs(Z - roots[0]) < tol
root2 = np.abs(Z - roots[1]) < tol
root3 = np.abs(Z - roots[2]) < tol
root4 = np.abs(Z - roots[3]) < tol
img = np.zeros([npts,npts,3])
img[root1] = [1,0,0]
img[root2] = [1,1,0]
img[root3] = [1,0,1]
img[root4] = [0,0,1]
plt.figure(figsize=(4,4))
plt.imshow(img)
Out[13]:
<matplotlib.image.AxesImage at 0xfc0320dcc0>
In [14]:
roots = np.array([complex(np.cos(2*np.pi*i/5),np.sin(2*np.pi*i/5)) for i in range(5)])
npts = 500
x = np.linspace(-1.,1.,npts)
y = np.linspace(-1.j,1.j,npts)
X,Y = np.meshgrid(x,y)
Z = X + Y
niters = 100
for i in range(niters):
    Z -= (Z**5 - 1)/(5*Z**4)
tol = .0001
root1 = np.abs(Z - roots[0]) < tol
root2 = np.abs(Z - roots[1]) < tol
root3 = np.abs(Z - roots[2]) < tol
root4 = np.abs(Z - roots[3]) < tol
root5 = np.abs(Z - roots[4]) < tol
img = np.zeros([npts,npts,3])
img[root1] = [1,0,0]
img[root2] = [0,1,0]
img[root3] = [1,0,1]
img[root4] = [0,0,1]
img[root5] = [0,1,1]
plt.imshow(img)
Out[14]:
<matplotlib.image.AxesImage at 0xfc03778518>
In [15]:
roots = np.array([complex(np.cos(2*np.pi*i/8),np.sin(2*np.pi*i/8)) for i in range(8)])
npts = 500
x = np.linspace(-1.,1.,npts)
y = np.linspace(-1.j,1.j,npts)
X,Y = np.meshgrid(x,y)
Z = X + Y
niters = 100
for i in range(niters):
    Z -= (Z**8 - 1)/(8*Z**7)
tol = .0001
root1 = np.abs(Z - roots[0]) < tol
root2 = np.abs(Z - roots[1]) < tol
root3 = np.abs(Z - roots[2]) < tol
root4 = np.abs(Z - roots[3]) < tol
root5 = np.abs(Z - roots[4]) < tol
root6 = np.abs(Z - roots[5]) < tol
root7 = np.abs(Z - roots[6]) < tol
root8 = np.abs(Z - roots[7]) < tol
img = np.zeros([npts,npts,3])
img[root1] = [1,0,0]
img[root2] = [0,1,0]
img[root3] = [1,0,1]
img[root4] = [0,0,1]
img[root5] = [0,1,1]
img[root6] = [0,0,0]
img[root7] = [1,1,1]
img[root8] - [1,1,0]
plt.imshow(img)
Out[15]:
<matplotlib.image.AxesImage at 0xfc0497df98>
In [ ]:

Report 5

Introduction

Differential equations provide us with a way of modeling an astounding amount of phenomena. All we need is the information about a system's current state and the way in which that state changes. The state may be characterized by just one number or by a set of numbers. For Report 5, we're going to develop a system of differential equations for a planet orbiting a star from scratch, first a single star and then for when stars are orbiting each other. With P representing the planet, this is what our system will look like: $$\vec{P} = \begin{bmatrix} x \\ y \\ v_x \\ v_y \end{bmatrix}, \dfrac{d\vec{P}}{dt} = \begin{bmatrix} v_x \\ v_y \\ -\dfrac{GMx}{(\sqrt{x^2 + y^2})^3} \\ -\dfrac{GMy}{(\sqrt{x^2 + y^2})^3} \end{bmatrix}$$ There's a few options for us to "solve" this equation numerically. Once the equations are solved, we'll be able to determine the planet's orbit by plotting its position over time and additionally animating the orbit. The methods we'll be using involve making very small changes in the direction of the derivative to solve our equations for position and velocity. For the sake of brevity, in this report will only be concerned with the Fourth-Order Runge-Kutta Method since I experimentally determined it to return the best accuracy as it accounts for the most error.

Tasks Stated & Approach

Exercise 1

For this exercise I just created I first created a function that takes initial values of P and from that continuously solves the equations for velocity and acceleration. This is for a one star system for a star with mass = 1.Then I created the function that implements the Range-Kutta Method to generate different solutions for P over time.

In [2]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
In [3]:
def dP_dt(P,t):
    G = 1.0
    M = 1.0
    x,y,vx,vy = P
    r3 = np.hypot(x,y)**3
    ax = -G*M*x/r3
    ay = -G*M*y/r3
    return np.array([vx,vy,ax,ay])
In [4]:
def rk4_path(tmax, h, P0):
    steps = int(tmax/h)
    path = np.empty((steps+1, 4))
    P = np.copy(P0)
    path[0] = P
    for i in range(steps):
        F1 = h*dP_dt(P, i*h)
        F2 = h*dP_dt(P + F1/2, (i+0.5)*h)
        F3 = h*dP_dt(P + F2/2, (i+0.5)*h)
        F4 = h*dP_dt(P + F3, (i+1)*h)
        P += (F1 + 2*F2 + 2*F3 + F4)/6.
        path[i+1] = P
    return path

Exercise 2

Now I had to plot the solutions for a long enough length of time that I could observe the orbits.

In [5]:
def plot_rk4(tmax, h, P0, standalone=True):
    path = rk4_path(tmax, h, P0)
    plt.plot(path[:,0], path[:,1])
    plt.axis('equal')
    plt.xlim(-1.25,1.25)
    plt.ylim(-1.25,1.25)
    plt.plot(0,0,'yo',ms=1)
In [ ]:
P0 = [1.,0.,0.,1.]
plot_rk4(100,.01,P0)

Now in order to see what the orbits will look like for a range of initial states of P, I create two figures each with 121 subplots. In the first, I plot a range of initial velocities versus with initial positions of x at 1 and y at 0. The orbits with variable velocity are mostly regular ellipses unless the position is too close to the sun. The same is true for the plots with variable starting position. Overall, these graphs aren't too exciting.

In [ ]:
vx = 11
vy = 11
nplot = 1
plt.figure(figsize=(12,8))
plt.suptitle('orbits of a planet in a one star system with x vs. y components of its initial velocity')
vals = np.linspace(-1.,1.,11)
for j in range(vy):
    for i in range(vx):
        plt.subplot(vx, vy, nplot, frameon=False)
        P0 = [1.,0.,vals[i],vals[j]]
        plot_rk4(100,.01,P0)
        plt.xticks([])
        plt.yticks([])
        plt.xlim(-3,3)
        plt.ylim(-3,3)
        nplot += 1
        if vals[i] == -1.:
            plt.ylabel(vals[j])
        if vals[j] == 1.:
            plt.xlabel(vals[i])
In [ ]:
x = 11
y = 11
nplot = 1
plt.figure(figsize=(12,8))
plt.suptitle("x vs. y coordinates for a planet's orbit in a one star system")
vals = np.linspace(-1.,1.,11)
for j in range(y):
    for i in range(x):
        plt.subplot(x, y, nplot, frameon=False)
        P0 = [vals[i],vals[j],.5,.5]
        plot_rk4(100,.01,P0)
        plt.xticks([])
        plt.yticks([])
        plt.xlim(-3,3)
        plt.ylim(-3,3)
        nplot += 1
        if vals[i] == -1.:
            plt.ylabel(vals[j])
        if vals[j] == 1.:
            plt.xlabel(vals[i])

Exercise 3/4

Now we want to solve for the state of P when the forces acting on it are affected by a 2-star system. To do this, we rewrite the dP_dt function to find the acceleration when there's 2 different sources of gravity acting upon it. This simply requires adding additional variables to reflect the doubling of forces, since we're finding values for the scalar components. For this report, we'll consider stars that are of the same mass. Then, we essentially use the exact same plot function but with the path of the Runge Kutta function and create a figure with similar conditions. I created the circle to represent the orbit of the 2 stars around their center of mass by adding a circle to the axes of the figure.

In [ ]:
def dP_dt2(P,t):
    G = 1.0
    M1 = .5
    M2 = .5
    R1 = .5
    R2 = .5
    x1 = R1*np.cos(t)
    x2 = -R2*np.cos(t)
    y1 = R1*np.sin(t)
    y2 = -R2*np.sin(t)
    x,y,vx,vy = P
    r3a = np.hypot(x-x1,y-y1)**3
    r3b = np.hypot(x-x2,y-y2)**3
    ax1 = -G*M1*(x-x1)/r3a
    ay1 = -G*M1*(y-y1)/r3a
    ax2 = -G*M2*(x-x2)/r3b
    ay2 = -G*M2*(y-y2)/r3b
    return np.array([vx,vy,ax1+ax2,ay1+ay2])


def rk4_path2(tmax,h,P0):
    steps = int(tmax/h)
    path = np.empty((steps+1,4))
    P = np.copy(P0)
    path[0] = P
    for i in range(steps):
        F1 = h*dP_dt2(P, i*h)
        F2 = h*dP_dt2(P + F1/2, (i+0.5)*h)
        F3 = h*dP_dt2(P + F2/2, (i+0.5)*h)
        F4 = h*dP_dt2(P + F3, (i+1)*h)
        P += (F1 + 2*F2 + 2*F3 + F4)/6.
        path[i+1] = P
    return path


def plot_rk42(tmax,h,P):
    path = rk4_path2(tmax,h,P0 )
    plt.plot(path[:,0],path[:,1])
    plt.axis('equal')
    plt.xlim(-1.,1.)
    plt.ylim(-1.,1.)
    fig = plt.gcf()
    ax = fig.gca()
    circle = plt.Circle((0,0),.5,color = 'y',fill=False)
    ax.add_artist(circle)

Below are subplots with variable initial velocity and a close fixed position to the orbit of the 2 suns to find a relatively stable orbit.

In [ ]:
plt.figure(figsize=(12,8))
plt.suptitle("orbits of a planet in a two star system with x vs. y components of its initial velocity")
vx = 11
vy = 11
nplot = 1
vals = np.linspace(-1.,1.,11)
for i in range(vy):
    for j in range(vx):
        plt.subplot(vx, vy, nplot, frameon=False)
        P0 = [.5000001,.5000001,vals[i],vals[j]]
        plot_rk42(100,.01,P0)
        plt.xticks([])
        plt.yticks([])
        nplot += 1
        if vals[i] == -1.:
            plt.ylabel(vals[j])
        if vals[j] == 1.:
            plt.xlabel(vals[i])
In [ ]:
plt.figure(figsize=(12,8))
plt.suptitle("x vs. y coordinates for a planet's orbit in a two star system")
x = 11
y = 11
nplot = 1
plt.figure(figsize=(12,8))
vals = np.linspace(-1.,1.,11)
for j in range(y):
    for i in range(x):
        plt.subplot(x, y, nplot, frameon=False)
        P0 = [vals[i],vals[j],0.,0.]
        plot_rk42(100,.01,P0)
        plt.xticks([])
        plt.yticks([])
        nplot += 1
        if vals[i] == -1.:
            plt.ylabel(vals[j])
        if vals[j] == 1.:
            plt.xlabel(vals[i])

All of the plots with variable x and y have orbits which end up diverging from regularity. However, when we examine cases close to the sun with variable velocity, some interesting orbits surface. Below are some zoomed in. I include a smaller h since there won't be a terrible impact on the runtime.

In [ ]:
P0 = [.500001,.500001,1.,-1.]
plot_rk42(100,.001,P0)
plt.xlim(-2.,2.)
plt.ylim(-2.,2.)

This plot is of particular note. It's quite beautiful, and could probably be described as a regular orbit. However, at first glance wouldn't necessarily hold any sort of significance. While I was working on this, a passing graduate student watched the orbit's animation and thought it might be a cross section of a torus, but the third circle inside the two overlapping didn't make sense. However, he then said that if it were viewed overhead, you could think of it as a path on some torus dependent on some $\theta$. He said that if $\theta$ is rational, then eventually the path would return to where it initially started. If it was irrational however, then the path would never return to the starting position and would never actually hit the same point twice. As we can see from the graphs below, as you increase the time from the start of the orbit, more new orbits keep forming. Thus it seems that if we treated this plot as a torus that the angle on the torus that we start from is irrational.

In [ ]:
P0 = [.500001,.500001,1.,-1.]
plot_rk42(100,.001,P0)
plt.xlim(-2.,2.)
plt.ylim(-2.,2.)
In [ ]:
P0 = [.500001,.500001,1.,-1.]
plot_rk42(300,.001,P0)
plt.xlim(-2.,2.)
plt.ylim(-2.,2.)
In [ ]:
P0 = [.500001,.500001,1.,-1.]
plot_rk42(400,.001,P0)
plt.xlim(-2.,2.)
plt.ylim(-2.,2.)
In [ ]:
P0 = [.5000001,.5000001,.6,.6]
plot_rk42(150,.01,P0)
plt.xlim(-2.,2.)
plt.ylim(-2.,2.)

That one is pretty. Any small changes one makes in the above position result in different snowflake-shaped orbits. I would even say that this is a regular orbit.

In [ ]:
P0 = [.5000001,.5000001,.8,.6]
plot_rk42(150,.01,P0)
plt.xlim(-2.,2.)
plt.ylim(-2.,2.)

I'd also classify the above as a regular orbit.

Exercise 6

These orbits can be animated. This is accomplished through the animate package of matplotlib. I also had to create a new path function that would give me the position of the stars at a certain time.

In [ ]:
def star_path(tmax,h):
    steps = int(tmax/h)
    R1 = .5
    R2 = .5
    path = np.empty((steps+1,4))
    for t in range(steps):
        x1 = R1*np.cos(t)
        x2 = -R2*np.cos(t)
        y1 = R1*np.sin(t)
        y2 = -R2*np.sin(t)
        path[t] = [x1,y1,x2,y2]
    return path
In [ ]:
%matplotlib
from matplotlib import animation
In [ ]:
tmax = 200
h = 0.01
P0 = np.array([.5000001,.5000001,1.,-1.])
path = rk4_path2(tmax, h, P0)
path2 = star_path(tmax, h)
frames = int(tmax/h)
fig, ax = plt.subplots(figsize=(8,8))
ax.set_aspect('equal')
ax.set_axis_bgcolor('k')
planet, = plt.plot([], [], 'bo')
orbit, = plt.plot([], [], 'cyan', lw=1)
star1, = plt.plot([], [], 'yo')
star2, = plt.plot([], [], 'yo')

width = .25 + np.amax(np.abs(path[:,:2]))
plt.xlim(-width, width)
plt.ylim(-width, width)

def animate(frame, path, planet, orbit, star1, star2):
    x, y = path[frame, :2]
    x1, y1 = path2[frame, :2]
    x2, y2 = path2[frame, 2:]
    planet.set_data(x, y)
    star1.set_data(x1,y1)
    star2.set_data(x2,y2)
    orbit.set_data(path[:frame, 0], path[:frame, 1])
    return orbit, planet, star1, star2

animation.FuncAnimation(fig, animate, frames=frames, fargs=(path, planet, orbit, star1, star2), interval=5, blit=True, repeat=True)
In [ ]:
P0 = np.array([.5000001,.5000001,.6,.6])
path = rk4_path2(tmax, h, P0)
path2 = star_path(tmax, h)
frames = int(tmax/h)
fig, ax = plt.subplots(figsize=(8,8))
ax.set_aspect('equal')
ax.set_axis_bgcolor('k')
planet, = plt.plot([], [], 'bo')
orbit, = plt.plot([], [], 'cyan', lw=1)
star1, = plt.plot([], [], 'yo')
star2, = plt.plot([], [], 'yo')
width = .25 + np.amax(np.abs(path[:,:2]))
plt.xlim(-width, width)
plt.ylim(-width, width)
animation.FuncAnimation(fig, animate, frames=frames, fargs=(path, planet, orbit, star1, star2), interval=5, blit=True, repeat=True)
In [ ]:
P0 = np.array([.5000001,.5000001,.8,.6])
path = rk4_path2(tmax, h, P0)
path2 = star_path(tmax, h)
frames = int(tmax/h)
fig, ax = plt.subplots(figsize=(8,8))
ax.set_aspect('equal')
ax.set_axis_bgcolor('k')
planet, = plt.plot([], [], 'bo')
orbit, = plt.plot([], [], 'cyan', lw=1)
star1, = plt.plot([], [], 'yo')
star2, = plt.plot([], [], 'yo')
width = .25 + np.amax(np.abs(path[:,:2]))
plt.xlim(-width, width)
plt.ylim(-width, width)
animation.FuncAnimation(fig, animate, frames=frames, fargs=(path, planet, orbit, star1, star2), interval=5, blit=True, repeat=True)

Exercise 7

The assumption that the two star's masses were equal was a pretty big one for real world settings. Modeling the the effects on planetary orbit due to stars of unequal masses is actually pretty easily done; it just requires a few changes to our original functions. Below I'm modeling for the Kepler binary star system. I found the masses by solvingthis system of equations: $$\frac{.94}{.20} = \frac{x_1}{x_2}$$ $$x_1 + x_2 = 1$$

In [ ]:
%matplotlib inline
def dP_dt2(P,t):
    G = 1.0
    M1 = .2703
    M2 = 0.7297
    R1 = .5
    R2 = .5
    x1 = R1*np.cos(t)
    x2 = -R2*np.cos(t)
    y1 = R1*np.sin(t)
    y2 = -R2*np.sin(t)
    x,y,vx,vy = P
    r3a = np.hypot(x-x1,y-y1)**3
    r3b = np.hypot(x-x2,y-y2)**3
    ax1 = -G*M1*(x-x1)/r3a
    ay1 = -G*M1*(y-y1)/r3a
    ax2 = -G*M2*(x-x2)/r3b
    ay2 = -G*M2*(y-y2)/r3b
    return np.array([vx,vy,ax1+ax2,ay1+ay2])


def rk4_path2(tmax,h,P0):
    steps = int(tmax/h)
    path = np.empty((steps+1,4))
    P = np.copy(P0)
    path[0] = P
    for i in range(steps):
        F1 = h*dP_dt2(P, i*h)
        F2 = h*dP_dt2(P + F1/2, (i+0.5)*h)
        F3 = h*dP_dt2(P + F2/2, (i+0.5)*h)
        F4 = h*dP_dt2(P + F3, (i+1)*h)
        P += (F1 + 2*F2 + 2*F3 + F4)/6.
        path[i+1] = P
    return path


def plot_rk42(tmax,h,P):
    path = rk4_path2(tmax,h,P0 )
    plt.plot(path[:,0],path[:,1])
    plt.axis('equal')
    plt.xlim(-1.,1.)
    plt.ylim(-1.,1.)
    fig = plt.gcf()
    ax = fig.gca()
    circle = plt.Circle((0,0),.5,color = 'y',fill=False)
    ax.add_artist(circle)
In [ ]:
vx = 11
vy = 11
nplot = 1
plt.figure(figsize=(12,8))
plt.suptitle('Orbits of the Kepler system with Planet starting at rest')
vals = np.linspace(-1.,1.,11)
for j in range(vy):
    for i in range(vx):
        plt.subplot(vx, vy, nplot, frameon=False)
        P0 = [.5000001,.5000001,vals[i],vals[j]]
        plot_rk42(300,.01,P0)
        plt.xticks([])
        plt.yticks([])
        nplot += 1
        if vals[i] == -1.:
            plt.ylabel(vals[j])
        if vals[j] == 1.:
            plt.xlabel(vals[i])

All in all, these orbits appear somewhat more chaotic, or at the least more likely to diverge than with stars of equal mass.

In [ ]:
P0 = [.5000001,.5000001,1.,-1.]
plot_rk42(100,.01,P0)
plt.xlim(-3.,3.)
plt.ylim(-3.,3.)
In [ ]:
P0 = [.5000001,.5000001,1.,-1.]
plot_rk42(500,.01,P0)
plt.xlim(-3.,3.)
plt.ylim(-3.,3.)
In [ ]:
P0 = [.5000001,.5000001,.9,-1.]
plot_rk42(100,.01,P0)
plt.xlim(-3.,3.)
plt.ylim(-3.,3.)
In [ ]:
P0 = [.5000001,.5000001,.9,-1.]
plot_rk42(500,.01,P0)
plt.xlim(-3.,3.)
plt.ylim(-3.,3.)

Report 6

Introduction

Report 6 is concerned with the methodology of generating random numbers and how they can be used to analytically determine solutions to problems that otherwise aren't easily solved. Through their use we can more easily observe trends of data over long periodss of time. They are applied to the natural sciences, statistics, game theory, finance, art, and much more.

But first: how do we generate random numbers? This will be done on a deterministic computer. Therefore it makes sense to use a deterministic algorithm which generates sequences of numbers that can be on some intervals viewed as "random". We'd also like it to be as simple as possible, with minimal "state." Don't wanna have to store and generate a ton of information.

Ideally, we want $x_{i+1} = g(x)$ where g is some function.

From there, it is now possible to generate normal distributions from the sums of random numbers. Using the matplotlib.pylot plt.hist() is what allows this data to be visualized.

Exercise 1

Linear congruential generators output random numbers in [0,1) with a sequence defined as follows:

$$x_{i} = a * x_{i-1} \text{ mod m}$$$$u_{i} = x_{i}/m$$

Below are the definitions of two functions, randu and rng, which are both LCG's. Randu is expected to return random numbers with less of a normal distribution than rng, just because rng uses better values for a and m which have since been found.

In [1]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from numpy.random import rand
from numpy.random import randint
from numpy.random import randn
In [2]:
def rng(x, npts):
    a = 427419669081
    m = 999999999989
    u = np.empty(npts)
    u[0] = x/m
    for i in range(1, npts):
        x = (a*x) % m
        u[i] = x/m
    return u
def randu(x, npts):
    a = 2**16 + 3
    m = 2**31
    u = np.empty(npts)
    u[0] = x/m
    for i in range(1, npts):
        x = (a*x) % m
        u[i] = x/m
    return u
In [3]:
rng_hist = rng(4000,10000)
randu_hist = randu(4000,10000)
plt.hist(rng_hist,bins=25)
plt.show()
plt.hist(randu_hist,bins=25)
plt.show()
plt.hist(rand(4000),bins=25)
plt.show()

It's fairly apparent from the above graphs that the NumPy rand function exhibits the most variance and non-uniformity, followed by rng and then randu. The average step size definitely appears to be larger for rng and rand, which is more desirable.

Exercise 2

Now that random numbers are at play we can apply them to equations. Just based upon the amount of points in a given area, we can estimate the radius of a circle inside of a square. To do this, we create random points on an xy-plane and find out how many are in the circle, which will give us the ratio of the circle's area to the square's. With a square of length 1 and circle of radius 1, the circle's area should end up being equivalent to pi. We should get an estimation within a tenth of a decimal place.

In [23]:
npts = 5000
x, y = rand(2, npts)*2 - 1
radius = 1
plt.figure(figsize=(8,8))
plt.plot(x, y, 'r.')
circle = plt.Circle((0, 0), radius, fill=False)
plt.gca().add_artist(circle)
plt.xlim(-1,1)
plt.ylim(-1,1)
pts_inside = 0
for i in range(npts):
    if np.hypot(x[i],y[i]) < 1:
        pts_inside += 1
pi = 4*pts_inside/npts
print(pi)

3.152

Exercise 3

Without doing any mental calculations, after thinking about it for a few seconds I would say more likely due to the fact that every point gained by Serena lessens Venus's chance to win.

To solve for Serena's probability of winning, we can just run create arrays of Serena's scores by testing if the elements of a 7-element random array are less than .6, equivalent to the likelihood of Serena scoring. Arrays which sum to four or more result in a win for Serena.

Results are pretty much as expected.

In [5]:
ngames = 1000
points = rand(7, ngames) < .6
scores = np.sum(points, axis=0)
wins = scores >= 4
plt.hist(scores,bins = 7,normed=True)
plt.title("Serena's final score")
plt.xlabel('points')
print('Serena has a',np.sum(wins)/ngames,'probability of winning')

Serena has a 0.696 probability of winning

Exercise 4

This is a pretty cut and dry exercise at first. Just add some random tuples and make a histogram of the sums.

In [6]:
plt.figure(figsize=(8,6))
npts = 10000
sums = np.sum(rand(2, npts), axis=0)
plt.hist(sums, bins=30, normed=True);

To test whether the sums of M-tuples converge in some way, we'll want to end up plotting them on the same graph. However, as evident below, the distribution is centered over increasing points. Thus, we want to move it back to a center of 0, and additionally manipulate the distributions to have the same bounds.

In [7]:
plt.figure(figsize=(15,3))
npts = 10000
for M in range(2, 7):
    sums = np.sum(rand(M, npts), axis=0)
    plt.subplot(1, 5, M-1)
    plt.hist(sums, bins=30, normed=True,ec='b')
    plt.xlim(0, 6)

So to center them all, we'll subtract M/2 from the sums when plotting the histogram.

In [8]:
plt.figure(figsize=(8,8))
npts = 10000
for M in range(2, 20):
    sums = np.sum(rand(M, npts), axis=0)
    plt.hist(sums-M/2, bins=50, normed=True, histtype='step')
    plt.xlim(-M/4, M/4)

The horizontal distance from each bounds decreases as M increases, leading me to believe that it could be modeled as some type of root or logarithmic function of M. Then, if we divided the sums of our random variables by that function we could potentially have convergence.

In [9]:
plt.figure(figsize=(8,8))
npts = 10000
for M in range(2, 20):
    sums = np.sum(rand(M, npts), axis=0)
    plt.hist((sums-M/2)/np.log(M), bins=50, normed=True, histtype='step')
    plt.xlim(-M/4/np.log(M), M/4/np.log(M))
In [10]:
plt.figure(figsize=(8,8))
npts = 10000
for M in range(2, 20):
    sums = np.sum(rand(M, npts), axis=0)
    plt.hist((sums-M/2)/M**.5, bins=50, normed=True, histtype='step')
    plt.xlim(-M/4/M**.5, M/4/M**.5)

Both converge very nicely to a normal distribution, especially the 2nd.

Exercise 5

Now we consider the probability density function for exponential distribution:

$$p(t) = \begin{cases} \lambda e^{-\lambda t} & t \geq 0 \\ 0 & t < 0 \end{cases}$$

Now we'll consider the radioactive decay of Carbon-14 to Nitrogen-14.

Below I created 10000 random numbers that I will use for my u, or in reality the antiderivative of the PDF such that $$\int^{g(u)}_{g(0)}p(t)dt = u$$

and then plug those values into my g(u) function which is really just u solved for in terms of t. These return the original time values that would have caused p(t) to return the random u values I generated. Now we measure the accuracy of generating the random u's by plotting their distribution against the PDF itself.

In [11]:
lam = 1/8267
def p(t): return lam*np.exp(-t*lam)
def g(u): return -np.log(1-u)/lam

u_values = rand(10000)
t_values = g(u_values)
t_values.sort()
plt.hist(t_values,bins=100,normed=True,histtype='step')
t = np.linspace(0,80000)
plt.plot(t,p(t),'r',lw=1.5)
plt.title('PDF of radioactive decay from Carbon-14 to Nitrogen-14')
Out[11]:
<matplotlib.text.Text at 0x9d0f9bacf8>

The generation of random time values lines up quite nicely.

Similar to in exercise 5 I plot the distributions of random M-tuples which are put into the g(u) function and then summed.

In [12]:
plt.figure(figsize=(8,15))
npts = 10000
for M in range(2, 50):
    random_times = rand(M, npts)
    us = g(random_times)
    sums = np.sum(us, axis=0)
    plt.hist(sums, bins=50, normed=True, histtype='step')
    plt.xlim(0,650000)
    plt.ylim(0,.00003)

Nothing too surprising here, it resembles an integral of the PDF.

Exercise 6

This classic math teaser astounded me when I was younger and it is thus extremely satisfying for me to actually analytically prove the results. I had already known of this problem before doing this report but when I first heard the problem I didn't understand how the choice to switch could possibly matter. However, now I realize that once a cup is removed, all of the probability that it would be a winning choice transfers to the other cup, rather than the odds equally distributing. This becomes very evident in the code.

In [13]:
trials = 100000
chosen = np.ones(ngames, dtype = int)
sum = 0
for i in range(trials):
    winning = randint(1,4,size = ngames)
    switched = np.where(winning == chosen, randint(2,4, size=ngames), winning)
    sum += np.mean(switched == winning)
print("The probability of winning after switching is",sum/100000)
print("The probability of winning without is thus",1-sum/100000)

The probability of winning after switching is 0.66668858
The probability of winning without is thus 0.33331142

Exercise 7

Now we want to use radn(), which returns a value based on that value's frequency in a normal distribution. We use that in our stochastic equation:

$$s(t + 1) = (1 + \mu + \sigma * \epsilon) * s(t)$$

Where epsilon is determined by randn(). As it turns out, although epsilon can be negative, the values of s will always remain positive; this makes sense, considering in a stock market a buyer will never have "negative" value, and wouldn't lose all of their money. I'm assuming that this equation is the solution to a partial differential equation.

Below are plots with varying epsilon (although it also changes for each day passed).

In [14]:
plt.figure(figsize=(20,5))
mu = 0.0
sigma = 0.05
s = np.empty(365)
s[0] = 1
for i in range(5):
    for t in range(364):
        s[t+1] = (1 + mu + sigma * randn()) * s[t]
    plt.subplot(1,5,i+1)
    plt.plot(s)

These plots do appear to display some uniformity in my view, with there being a somewhat predictable frequency of sharp critical points along with bunches of closer ones. It's obviously very random, but there could be some order to it.

Below is a plot of the frequency

In [15]:
mu = 0.0
sigma = 0.05
days = 365
ntrials = 10000
s = np.empty(days+1)
s[0] = 1
finals = np.zeros(ntrials)
for i in range(ntrials):
    for t in range(days):
        s[t+1] = (1 + mu + sigma * randn()) * s[t]
    finals[i] = s[days]
    plt.plot(s)

Plotting 1000 different stock price changes for the same values of mu and epsilon produced similar trajectories. They don't seem to converge to anything but definitely cluster to some value less than 0. Interestingly, there can be some paths which grow much higher than value in others and don't appear to yet be dwindling. I learned that this is called a "stochastic bubble".

In [16]:
plt.hist(finals,bins=100)
plt.xlim(0,5)
Out[16]:
(0, 5)

Based upon observations from the last plot, the fact that the median of final values is below 0 is not surprising.

The average of the stock's worth one year after purchasing for one unit of money is

In [17]:
np.average(finals)
Out[17]:
1.0100877748860442

The likelihood of losing money is

In [18]:
np.average(finals < 1)
Out[18]:
0.68059999999999998

I'm guessing that the best guess for most likely value would be the median, since it is the most likely to occur in a random sampling.

In [19]:
np.median(finals)
Out[19]:
0.64913844669463705

The average can actually be over 1; however, the probability of lower returns is greater than that of higher. This is caused by outliers which finished with extremely high returns skewing the results. The returns can only be 1 less than the start, but any amount higher governed by some amount of time. The median is also surprisingly low; the fact that half off all returns are less than that makes it evident that playing this specific stock market is very risky. I personally would not trade in such a stock.

For a value for sigma of .01, things very much change. It is still more likely that you'll be shorted in the end, but by less of a degree.

In [20]:
mu = 0.0
sigma = 0.01
days = 365
ntrials = 1000
s = np.empty(days+1)
s[0] = 1
prices = np.zeros((days,ntrials))
finals = np.zeros(ntrials)
for i in range(ntrials):
    for t in range(days):
        s[t+1] = (1 + mu + sigma * randn()) * s[t]
        prices[t,i] = s[t]
    finals[i] = s[days]
plt.hist(prices,bins=200,normed=True,histtype='step')

C:\Users\John\Anaconda3\lib\site-packages\matplotlib\axes\_axes.py:5882: UserWarning: 2D hist input should be nsamples x nvariables;
 this looks transposed (shape is 365 x 1000)
  '(shape is %d x %d)' % inp.shape[::-1])
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           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ,
           0.        ,   0.        ,   0.        ,   0.        ])],
 array([ 0.5259728 ,  0.53229866,  0.53862452,  0.54495038,  0.55127624,
         0.5576021 ,  0.56392796,  0.57025382,  0.57657968,  0.58290555,
         0.58923141,  0.59555727,  0.60188313,  0.60820899,  0.61453485,
         0.62086071,  0.62718657,  0.63351243,  0.6398383 ,  0.64616416,
         0.65249002,  0.65881588,  0.66514174,  0.6714676 ,  0.67779346,
         0.68411932,  0.69044518,  0.69677104,  0.70309691,  0.70942277,
         0.71574863,  0.72207449,  0.72840035,  0.73472621,  0.74105207,
         0.74737793,  0.75370379,  0.76002966,  0.76635552,  0.77268138,
         0.77900724,  0.7853331 ,  0.79165896,  0.79798482,  0.80431068,
         0.81063654,  0.8169624 ,  0.82328827,  0.82961413,  0.83593999,
         0.84226585,  0.84859171,  0.85491757,  0.86124343,  0.86756929,
         0.87389515,  0.88022102,  0.88654688,  0.89287274,  0.8991986 ,
         0.90552446,  0.91185032,  0.91817618,  0.92450204,  0.9308279 ,
         0.93715377,  0.94347963,  0.94980549,  0.95613135,  0.96245721,
         0.96878307,  0.97510893,  0.98143479,  0.98776065,  0.99408651,
         1.00041238,  1.00673824,  1.0130641 ,  1.01938996,  1.02571582,
         1.03204168,  1.03836754,  1.0446934 ,  1.05101926,  1.05734513,
         1.06367099,  1.06999685,  1.07632271,  1.08264857,  1.08897443,
         1.09530029,  1.10162615,  1.10795201,  1.11427788,  1.12060374,
         1.1269296 ,  1.13325546,  1.13958132,  1.14590718,  1.15223304,
         1.1585589 ,  1.16488476,  1.17121062,  1.17753649,  1.18386235,
         1.19018821,  1.19651407,  1.20283993,  1.20916579,  1.21549165,
         1.22181751,  1.22814337,  1.23446924,  1.2407951 ,  1.24712096,
         1.25344682,  1.25977268,  1.26609854,  1.2724244 ,  1.27875026,
         1.28507612,  1.29140198,  1.29772785,  1.30405371,  1.31037957,
         1.31670543,  1.32303129,  1.32935715,  1.33568301,  1.34200887,
         1.34833473,  1.3546606 ,  1.36098646,  1.36731232,  1.37363818,
         1.37996404,  1.3862899 ,  1.39261576,  1.39894162,  1.40526748,
         1.41159335,  1.41791921,  1.42424507,  1.43057093,  1.43689679,
         1.44322265,  1.44954851,  1.45587437,  1.46220023,  1.46852609,
         1.47485196,  1.48117782,  1.48750368,  1.49382954,  1.5001554 ,
         1.50648126,  1.51280712,  1.51913298,  1.52545884,  1.53178471,
         1.53811057,  1.54443643,  1.55076229,  1.55708815,  1.56341401,
         1.56973987,  1.57606573,  1.58239159,  1.58871745,  1.59504332,
         1.60136918,  1.60769504,  1.6140209 ,  1.62034676,  1.62667262,
         1.63299848,  1.63932434,  1.6456502 ,  1.65197607,  1.65830193,
         1.66462779,  1.67095365,  1.67727951,  1.68360537,  1.68993123,
         1.69625709,  1.70258295,  1.70890882,  1.71523468,  1.72156054,
         1.7278864 ,  1.73421226,  1.74053812,  1.74686398,  1.75318984,
         1.7595157 ,  1.76584156,  1.77216743,  1.77849329,  1.78481915,
         1.79114501]),
 <a list of 1000 Lists of Patches objects>)
In [21]:
plt.hist(finals,bins=100)
Out[21]:
(array([  1.,   2.,   1.,   3.,   2.,   0.,   5.,   2.,   2.,   7.,   4.,
          5.,   5.,  10.,   9.,   9.,  17.,  24.,  16.,  20.,  19.,  12.,
         18.,  24.,  21.,  22.,  31.,  24.,  30.,  33.,  29.,  27.,  23.,
         27.,  12.,  30.,  27.,  31.,  18.,  15.,  16.,  28.,  33.,  23.,
         21.,  19.,  24.,  17.,  19.,  10.,  17.,  20.,  13.,   6.,   9.,
         10.,   5.,  13.,  10.,   7.,   7.,   4.,   7.,   3.,   3.,   3.,
          6.,   2.,   3.,   2.,   2.,   2.,   1.,   1.,   0.,   2.,   2.,
          2.,   2.,   1.,   1.,   1.,   0.,   1.,   0.,   0.,   1.,   1.,
          0.,   0.,   1.,   0.,   1.,   0.,   0.,   0.,   0.,   0.,   0.,
          1.]),
 array([ 0.53607859,  0.54855492,  0.56103125,  0.57350757,  0.5859839 ,
         0.59846023,  0.61093656,  0.62341288,  0.63588921,  0.64836554,
         0.66084187,  0.67331819,  0.68579452,  0.69827085,  0.71074718,
         0.72322351,  0.73569983,  0.74817616,  0.76065249,  0.77312882,
         0.78560514,  0.79808147,  0.8105578 ,  0.82303413,  0.83551045,
         0.84798678,  0.86046311,  0.87293944,  0.88541576,  0.89789209,
         0.91036842,  0.92284475,  0.93532107,  0.9477974 ,  0.96027373,
         0.97275006,  0.98522638,  0.99770271,  1.01017904,  1.02265537,
         1.03513169,  1.04760802,  1.06008435,  1.07256068,  1.08503701,
         1.09751333,  1.10998966,  1.12246599,  1.13494232,  1.14741864,
         1.15989497,  1.1723713 ,  1.18484763,  1.19732395,  1.20980028,
         1.22227661,  1.23475294,  1.24722926,  1.25970559,  1.27218192,
         1.28465825,  1.29713457,  1.3096109 ,  1.32208723,  1.33456356,
         1.34703988,  1.35951621,  1.37199254,  1.38446887,  1.39694519,
         1.40942152,  1.42189785,  1.43437418,  1.44685051,  1.45932683,
         1.47180316,  1.48427949,  1.49675582,  1.50923214,  1.52170847,
         1.5341848 ,  1.54666113,  1.55913745,  1.57161378,  1.58409011,
         1.59656644,  1.60904276,  1.62151909,  1.63399542,  1.64647175,
         1.65894807,  1.6714244 ,  1.68390073,  1.69637706,  1.70885338,
         1.72132971,  1.73380604,  1.74628237,  1.75875869,  1.77123502,
         1.78371135]),
 <a list of 100 Patch objects>)
In [22]:
print(np.average(finals),np.average(finals < 1),np.median(finals))

0.986763493202 0.555 0.974576093834
In [ ]: